differentiate implicitly to find \\( \\frac { d y } { d x } \\)\n\n\\( y ^ { 4 } = x ^ { 9 } \\)\n\n\\(…

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (y^{4}) using the chain - rule. The derivative of (u^{n}) with respect to (x) is (n\cdot u^{n - 1}\cdot\frac{du}{dx}), where (u = y) and (n = 4). The derivative of (x^{9}) with respect to (x) is (9x^{8}) using the power - rule (\frac{d}{dx}(x^{n})=nx^{n - 1}). So, (\frac{d}{dx}(y^{4})=\frac{d}{dx}(x^{9})) gives (4y^{3}\frac{dy}{dx}=9x^{8}).

Step2: Solve for (\frac{dy}{dx})

Divide both sides of the equation (4y^{3}\frac{dy}{dx}=9x^{8}) by (4y^{3}). (\frac{dy}{dx}=\frac{9x^{8}}{4y^{3}})

Answer:

(\frac{9x^{8}}{4y^{3}})